Euclid's lemma
Okay kiddo, so you know when we count with numbers like 1, 2, 3, and 4? Well, sometimes we need to know if one of those numbers can be divided evenly by another number. For example, we know that 4 can be divided evenly by 2 because 4 is like having 2 groups of 2.
Euclid's lemma is like a rule that tells us if a number can be divided evenly by another number. The rule says that if a prime number (which is a number that can only be divided evenly by itself and 1) divides a product of two other numbers, then it must divide at least one of those numbers. That's a bit of a mouthful, so let's break it down.
First, we need to know what a prime number is. Do you remember when we talked about how some numbers can only be divided by themselves and 1? Those are prime numbers. Examples of prime numbers are 2, 3, 5, 7, 11, and so on.
Next, let's talk about products. When we multiply two numbers together, we get a product. For example, 2 times 5 equals 10, so 10 is the product of 2 and 5.
Now, let's put it all together. If we have a prime number like 2, and we multiply it by two other numbers, say 3 and 4 to get 2 times 3 times 4, which equals 24, then Euclid's lemma tells us that if 2 divides 24, then it must also divide either 3 or 4. In this case, 2 divides 4 evenly, so it also divides 24 evenly.
The reason why this lemma is helpful is that it can save us time when we need to figure out if a number can be divided evenly by another number. We can use this rule to quickly check if a prime number divides a product of two numbers, instead of having to check each number individually.
I hope that helps!
Euclid's lemma is like a rule that tells us if a number can be divided evenly by another number. The rule says that if a prime number (which is a number that can only be divided evenly by itself and 1) divides a product of two other numbers, then it must divide at least one of those numbers. That's a bit of a mouthful, so let's break it down.
First, we need to know what a prime number is. Do you remember when we talked about how some numbers can only be divided by themselves and 1? Those are prime numbers. Examples of prime numbers are 2, 3, 5, 7, 11, and so on.
Next, let's talk about products. When we multiply two numbers together, we get a product. For example, 2 times 5 equals 10, so 10 is the product of 2 and 5.
Now, let's put it all together. If we have a prime number like 2, and we multiply it by two other numbers, say 3 and 4 to get 2 times 3 times 4, which equals 24, then Euclid's lemma tells us that if 2 divides 24, then it must also divide either 3 or 4. In this case, 2 divides 4 evenly, so it also divides 24 evenly.
The reason why this lemma is helpful is that it can save us time when we need to figure out if a number can be divided evenly by another number. We can use this rule to quickly check if a prime number divides a product of two numbers, instead of having to check each number individually.
I hope that helps!
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